DOA ESTIMATION FOR FMCW RADAR BY 3D-CNN - ARTICLE - PREPRINTS.ORG

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 Article

 DoA Estimation for FMCW Radar by 3D-CNN
 Tzu-Hsien Sang 1,*, Feng-Tsun Chien 1, Chia-Chih Chang 1, and Kuan-Yu Tseng 1, Bo-Sheng Wang 2, Jiun-In Guo 1

 1 Institute of Electronics, National Yang Ming Chiao Tung University
 2 Institute of Electronics, National Chiao Tung University
 * Correspondence: tzuhsien54120@faculty.nctu.edu.tw

 Abstract: A method of direction-of-arrival (DoA) estimation for FMCW radars is presented. In ad-
 dition to MUSIC and ESPRIT, which are well-known high-resolution DoA estimation algorithms,
 deep learning has recently emerged as a very promising alternative. It is proposed in this paper to
 use a 3D convolutional neural network (CNN) for DoA estimation. The 3D-CNN extracts from the
 radar data cube spectrum features of the region of interest (RoI) centered on the potential positions
 of the targets, thereby capturing the spectrum phase shift information, which corresponds to DoA,
 along the antenna axis. Finally, the results of simulations and field experiments are provided to
 demonstrate the superior performance, as well as the limitations, of the proposed 3D-CNN.

 Keywords: FMCW Radar; Deep Learning; Three-Dimension Convolution Network; Direction-of-
 Arrival Estimation

 1. Introduction
 Radar is considered to be one of the key sensor technologies for enabling autonomous
 driving [1]; in particular mmW radar becomes very popular because it has advantages,
 especially compared to LiDAR (Light Detection And Ranging), such as small size, low
 transmit power, excellent range resolution, low clutter sensitivity, as well as it is relatively
 inexpensive while maintaining certain performance under almost all weather conditions
 [2,3]. The most commonly used waveform in low-cost mmW radar today is Frequency
 Modulated Continuous Wave (FMCW). By utilizing the large bandwidth available and
 the high carrier frequencies in automotive radar bands, good resolution can be achieved
 in range and speed. However, the angle resolution is limited by the number of antennas
 of the phased-array. Since the automotive radar sensor is a mass-produced product, cost
 is an important consideration, so the number of receiving antennas is usually low in the
 tens [4, 5].
 High-resolution DoA estimation has recently become very relevant in the current
 filed of autonomous driving and advanced driver assistant system (ADAS). Conventional
 high-resolution DoA estimation algorithms, such as MUSIC and ESPRIT [6-9], are based
 on sub-space methods. In the application of automotive radars, usually only a few num-
 bers of measurement snapshots can be used. However, the sub-space methods can only
 achieve the best performance when enough snapshots are available. And because they are
 based on presumed signal models, the performance of these algorithms degrade fast when
 the signal-to-noise ratio (SNR) becomes low. In the scenarios typical in vehicular trans-
 portation, accurate DoA estimation becomes challenging [10, 11]. In addition, when the
 number of detected targets is unknown, and multiple targets have different radar cross-
 section (RCS), their performance may also be affected. The correct detection of the number
 of targets in the receiver signal is also an important issue in array signal processing [12].
 The aim of this paper is to present a 3D-CNN for high-resolution DoA estimation for
 FMCW radar. Up to two targets will be distinguished and their DoAs be estimated. Region
 of interest (RoI) is used to manage cases with more targets. Two formats of data are used
 in the process: the raw radar data cube and the statistical average in the form of covariance
 matrices. Simulation and experiment results are provided to demonstrate the superior

 © 2021 by the author(s). Distributed under a Creative Commons CC BY license.

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 performance of the proposed method over conventional methods such MUSIC as well as
 emerging deep learning methods [13-20]. Neural networks (NNs) have been shown more
 effective than MUSIC for estimating DoAs, for instance, when SNRs are relative low. But
 either no convincing experiment results have been shown to support the claims [14-20],
 or the angular differences between targets are not small enough to show the supremacy
 of deep-learning-based methods [20]. In this paper, experiment results in addition to sim-
 ulation results are presented to demonstrate the superior performance of the proposed
 3D-CNN. To overcome the issue of lacking the diversity of range in the training data set,
 a data augmentation technique is also developed. For scenarios of two targets with small
 angular differences, our 3D-CNN shows very promising results of resolving the angles
 for these critical cases. In addition, it is relatively easy to extend the proposed approach
 for processing the data cube to obtain the target’s other parameters in the future.

 2. Materials and Methods
 The task of estimating DoAs can be divided mainly into two stages, the first being
 signal preprocessing and the second being DoA estimation by CNN. The two stages will
 be described in the following two subsections. Furthermore, details of data generation and
 data augmentation in simulation/experiment are also provided.

 2.1. Signal Model and Signal Preprocessing
 In this paper the conventional model of plane waves impinging a Uniform Linear
 Array (ULA) is adopted [21]. Suppose that plane waves reflected from targets with
 complex amplitudes ( ) and wavelength λ come from directions , = 1, ⋯ , . The
 ULA contains antennas spaced by . After demodulation, the received signal of the
 ULA at snapshot time can be formulated as
 ( ) = ( ) + ( ), (1)

 where ( ), (t), and ( ) are
 ( ) = [ 1 ( ), … , ( )] , ( ) = [ 1 ( ), … , ( )] , ( ) = [ 1 ( ), … , ( )] . (2)

 In the above equations, ( ) (an × 1 vector) denotes the reflected signals coming from
 the far-field DoAs 1 , … , , and ( ) is an × 1 vector of independent additive white
 Gaussian noise (AWGN) with zero mean and variance of 2.
 Define an M×N matrix whose columns are steering vectors toward DoAs:
 = [ ( 1 ), ⋯ , ( )], (3)

 with
 
 ( ) = [1, − , − 2 , ⋯ , − ( −1) ] , (4)

 where ( ) is the steering vector and = 2 sin( )⁄λ is the phase difference be-
 tween adjacent elements. The demodulated signals contain enough information to esti-
 mate the DoAs.
 Many DoA estimation methods use the × covariance matrix of ( ):
 = E[ ( ) ( )] = + , (5)

 where [. ]H and E[.] denotes the conjugate transpose and expectation operator. and
 are the × signal and noise covariance matrices, respectively. It is assumed that
 noise components are AWGN and the covariance matrix is
 = 2 , (6)

 where is the identity matrix and 2 is the noise power. In practice, is unknown, and
 the number of sampled signal vectors is limited. The maximum likelihood estimation of
 the spatial covariance matrix can be obtained from snapshots by :

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 1 
 ̂ = ∑ =1 ( ) ( ) . (7)
 
 The demodulated signal vectors contain sufficient information for DoA estimation.
 However, the size of data vectors can be huge that the training of NNs is greatly compli-
 cated. In radar applications, the signals reflected from multiple targets arrive at the re-
 ceiver at different times. It is desirable to use this fact to first do a coarse separation of
 target signals such that the training can be simplified. Therefore, RoI will be first estab-
 lished to limit the coverage of the NN, and target signals with very close ranges and DoAs
 in the range-azimuth spectrogram will be grouped into the same RoI for further pro-
 cessing
 FMCW radar is chosen to further build the signal model due to its popularity in ve-
 hicular applications [1,2], especially the signal framework of TI mmW radar
 AWR1843BOOST+DCA100EVM is adopted [22]. The process of generating data cube [23]
 and selecting RoI in the proposed scheme is summarized in Fig. 1 and 2. First, multiple
 snapshots are collected and each is transformed by DFT to generate range information.
 Afterwards all snapshots are stacked along the antenna and chirp axis to form a data cube
 for further processing. The selection of RoI is done by first detecting possible targets on
 the range axis by using the cell-averaging constant false alarm rate (CA-CFAR) principle
 [24-26]. Then the samples centered at the target will be claimed as RoI. If needed, a
 coarse DoA/speed interval can also be obtained by conventional DoA/speed estimation
 methods such as angle/speed DFT along the antenna/chirp axis [23]. The three-dimen-
 sional RoI of the data cube can be then cropped and further manipulated as the input of
 the deep learning framework.

 Figure 1. Generation of a data cube.

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 Figure 2. RoI selection in a data cube.

 The data cube is complex-valued. The NN will take the absolute value, the real and
 the imaginary part of the data as the input for DoA estimation. A similar data format has
 been found to be effective for training neural networks [27]. In addition to the RoI of the
 data cube, the covariance matrix is also used as the input for determining the number of
 targets. The details will be given in the next subsection. In order to reduce the impact from
 noise outside the RoI, the covariance matrix is estimated with only the signal within the
 RoI. As shown in Fig. 3, the data is first filtered by a band-pass filter fitted to the RoI, then
 the maximum likelihood estimation ̂ in (7) is obtained. Since ̂ is Hermitian, the di-
 agonal terms and upper triangular part of it contains all the information. Let ̂ be first
 normalized so that its real and imaginary parts are within [−1,1] and then be morphed
 into a vector ∈ RM(M−1)×1 for the input to a NN:
 ̂ 12 ),
 = [ (R ̂ 12 ), ⋯ , (R
 (R ̂ 1M ), ⋯ , (R
 ̂ 23 ), ⋯ , (R
 ̂ ( −1) )] (8)

 Figure 3. The covariance matrix of RoI.

 2.2. DoA Estimation by CNN
 Deep learning has breached into many application areas where conventional signal
 processing techniques are facing difficulties; among numerous neural network models,
 CNNs are known to be very capable of extracting features from matrix- or tensor-formed
 data [28-31]. Its architecture typically consists of one or more convolution layers followed
 by fully-connected networks leading to the output. In the proposed scheme, because a 3-
 D data cube is processed to make high-precision DoA estimation, a 3D-CNN [30] is de-
 ployed. But before the 3D-CNN, a simple fully-connected NN is first used for determining
 the number of targets. The overall signal flow together with the signal preprocessing is
 summarized in Fig. 4, and more details are itemized in the following.

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 Figure 4. Signal processing and deep learning architecture.

 1. Signal preprocessing and the selection of RoI have been described in the previous
 subsection.
 2. An independent small-size fully –connected NN is used to estimate the number of
 targets. The signal’s statistical average is found to be more robust to provide coarse
 information, such as the number of targets; while the data cube contains subtle infor-
 mation without being averaged out so that it enables high-resolution DoA estima-
 tion. The input of the simple fully-connected NN to estimate the number of targets is
 an × covariance matrix (in our case, = 8), and the covariance matrix is ob-
 tained by (8). The estimated number (0, 1, or 2) will then be used to choose the one-
 target or two-target 3D-CNN for estimating the DoAs.
 3. The 3D-CNN does DoA estimation by treating it as a classification task [32]. There
 are twenty one classes (corresponding to DoAs −10° , −9° , ⋯ , 10° ) to be classified. In
 Fig. 4, the highlighted RoI is the input to the 3D-CNN and it is a × × × 3 tensor
 (in our case, = 65, = 8, = 4, which stand for number of fast-time samples, of
 antennas, and of chirps respectively). The 3 is the number of data formats that in-
 cludes the absolute value, the real part, and imaginary part; using the arrangement
 is shown to yield better performance than only using the real and imaginary part
 [27]. The first convolutional layer in the 3-D CNN (Conv1) uses 30×5×3 kernels with
 the stride step of 1×1×1, and the number of channels is 256. The same number of
 channels are passed to the second convolutional layer (Conv2), which uses 2×2×2
 kernels, and a max pooling (MP) layer follows. The output of feature extraction is
 then flattened and connected into the form of a matrix, and a four-layer fully con-
 nected layer is used for its classification. The loss function used for the classification
 task is binary cross entropy in Pytorch environment [33].
 4. The parameters for the fully-connected NN and 3D-CNN are listed in Table 1 and 2.
 The term FC stands for Fully Connected and BN for Batch Normalization [34]. In the

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 training of this fully-connect NN and the 3D-CNN, the popular optimization tool
 ADAM and the drop-out technique are used [35, 36].

 Table 1. Fully-connected NN parameters.

 Layer # of Filter Activation Layer # of Filter Activation
 Input 56 FC3+BN 1024 ReLu
 FC1+BN 1024 ReLu FC4+BN 1024 ReLu
 FC2+BN 1024 ReLu FC5+BN 3 Sigmoid

 Table 2. 3D-CNN parameters.

 Layer Filter # of filter Activation Output
 RoI spectrum 65× 8 ×4×3
 Conv1 (30,5,3) 256 ReLu
 Conv2 (2,2,2) 256 ReLu
 MaxPool 1 (2,2,2)
 FC ReLu 500
 FC ReLu 200
 FC ReLu 100
 FC Sigmoid 21

 2.3. Data Generation and Data Augmentation
 Both simulation data and experiment data are acquired for training the NNs and
 evaluating the proposed DoA estimation scheme. The simulation data is primarily for
 evaluating and comparing the performance of DoA estimation with the Cramer-Rao
 Bound (CRB) [37] and existing algorithms for a wide range of SNRs. The experiment data,
 on the other hand, is primarily for demonstrating the applicability of the proposed DoA
 estimation scheme.
 Simulation data is generated by the MATLAB Phased Array tool box [38] with the
 parameters delineated below. To simplify the performance analysis, the number of targets
 is set to be 0, 1, or 2. The radar is an FMCW one with 8 receiving antennas. The resonance
 frequency is 77 GHz and one snapshot received at each antenna has = 256 sample
 points. The DoA of each incident plane wave ranges from -10° to 10°, and signals under
 various SNRs ranging from -10 to 15 dB are generated. Detailed simulation parameters
 are summarized in Table 3. In total, 63k snapshots are generated for training and 2.1k for
 each testing in different SNR. Note that all the pairs of DoA ([-10°,-9°], [-10°,-8°], …, [9°,10°])
 are considered in both training and testing.
 In simulation, the model is first trained by the first training data set with SNRs in the
 range from 15 to 0 dB; then the model is further strengthened by the second training data
 set with SNRs from 15 to -5 dB. Number of the both training data is 63k. The reason for
 doing the two-stage training is to make sure the model first learns the correct signal fea-
 tures with relatively high-SNR data then make the model more robust by training it with
 more noisy data.

 Table 3. Simulation parameters.

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 Parameters Value Parameters Value
 Transmit signal FMCW Target Number 0, 1, or 2
 Carrier Frequency 77 GHz Target Range 70 m to 90 m
 Frequency Slope 3.476 THz/s Target Angle -10° to 10°
 Bandwidth 0.15 GHz Samples per chirp 256
 Tx/Rx Antenna 1/8 SNR -10 dB to 15 dB

 Experiments with metal corner reflectors mounted on tripods on a baseball field
 (please see Fig. 5) are conducted to collect data. The TI mmW radar
 AWR1843BOOST+DCA100EVM is used [39]; it has 2 Tx and 4 Rx antennas to constitute a
 virtual 8-antenna receiving array. Detailed experiment parameters are summarized in Ta-
 ble 4. Note that some radar parameters used in simulation and experiments are not exactly
 the same; it is due to the limitations of both the MATLAB Phased Array tool box and the
 TI radar platform. The training data set for experiment includes both 21k raw snapshots
 (directly read from the TI radar) and 28k snapshots with data augmentation, and about
 7k raw snapshots are reserved for testing. Details and methods of data augmentation are
 provided below. The reason of using augmented data is to increase the robustness of our
 trained model.

 Figure 5. Experiments with two metal corner reflectors.

 Table 4. Experiment parameters.

 Parameters Value Parameters Value
 Transmit signal FMCW Target Number 0, 1, or 2
 Carrier Frequency 77 GHz Target Range 10 m
 Frequency Slope 4.2 THz/s Range Augmentation 9.7 m to 10.3 m
 Bandwidth 3.5 GHz Target Angle -10° to 10°
 Tx/Rx Antenna 2/4 Samples per chirp 512

 In addition to the data collected in the field experiment, data augmentation is de-
 ployed to increase the quantity and enhance the range diversity of the training data set.
 We suspect that the lack of reporting high-resolution results with using NNs on experi-
 ment data in literature might be just due to the robustness issue caused by insufficient
 range diversity. The experiments are conducted with the targets placed only at certain
 ranges. Nonetheless, real-world targets come with random ranges. Therefore, it is essen-
 tial to expose the NNs to targets with a variety of ranges. One way to do it without trying

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 to place targets at a multitude of ranges is implementing data augmentation. Note that
 the demodulated FMCW signal along the fast-time axis (in a data cube shown in Figure 2)
 exhibits a beat frequency corresponding to the target’s range:
 2 2 
 ) +
 ( ) = 2 [( 
 ] (9)

 where ( ) is the demodulated FMCW chirp signal, is the target’s range, is the car-
 rier frequency, is the bandwidth, is the chirp duration, and c is the speed of light.
 Therefore, it only takes a proper frequency term to change the target’s range. As a result,
 we only need to multiply a phase term to signals along the fast-time axis to create virtual
 targets at the range + ∆ , where ∆ is randomly chosen from -0.3 to 0.3 m:
 2 ∆ 2 ∆ 
 ) + ]
 aug ( ) = 2 [( × ( ). (10)

 From each angle set, there are several snapshots chosen to be augmented, and 220
 different ∆ ’s are generated to augment the size of data set to about 28k.

 3. Results
 In this section, the simulation results of evaluating and comparing the performance
 of the proposed DoA estimation method with the Cramer-Rao Bound (CRB) [37] and the
 DNN (Deep NN) in [19], which has the state-of-the-art performance among deep-learn-
 ing-based method methods, is first presented. Then the experiment results with up to two
 corner reflectors are presented to demonstrate the potential of the proposed method in
 applying to real-world situations. Note that the proposed DoA estimation method con-
 sists of mainly two steps: the first is to determine with a fully-connected NN the number
 of targets in the RoI, while the second is to estimate the targets’ DoAs with a 3D-CNN
 designed either for one target or for two targets. Therefore, for both simulation and ex-
 periment, the accuracy of estimating the number of targets is first characterized, followed
 by the results on DoA estimation.

 3.1. Simulation Results
 Fig. 6 shows the accuracy of determining the number of targets. As can be seen, the
 accuracy reaches 100% once the SNR hits 5 dB, and it almost maintains 90% even when
 the SNR is as low as -10 dB. The highly effective fully-connected NN provides a very en-
 suring condition for the 3D-CNN to work on DoA estimation.

 Figure 6. The accuracy of determining the number of targets vs. SNR.

 Fig. 7 presents the performance of DoA estimation for the case of one target. The root-
 mean-square error (RMSE) of the proposed method, MUSIC, the DNN in [19], and the
 CRB are compared. It is clear that both NN methods beat MUSIC and stay close to the

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 CRB across a wide range of SNRs. The proposed 3D-CNN and the DNN in [19] have sim-
 ilar performance and both perform better than MUSIC in most SNRs. The advantage of
 the 3D-CNN becomes evident in the case of two targets, as will be shown next.

 Figure 7. The RMSE of DoA estimation vs. SNR.

 Fig. 8 compares the success rate of obtaining correct DoAs for two-target cases. The
 success is defined with a ±1° estimation error. Four methods are compared. The original
 DNN proposed in [19] has about 0.135 million parameters, e.g. it is a very small DNN,
 especially compared to our 3D-CNN. It is no surprise that 3D-CNN has a clear advantage
 over the whole range of SNRs. However, the performance of the DNN improves when its
 size is increased. Therefore, we increase its size until the performance tops (after that large
 DNNs perform poorer due to difficulties in training), and the result is a large DNN with
 around 11 million parameters. As can been from Fig. 8, the 3D-CNN still has the best
 overall performance, even though the large DNN has diminished the performance gap to
 3 dB. It is also worth noting that at SNR = -10 dB, the 3D-CNN performs worse because
 the training data set does not contain data from this SNR. It shows that 3D-CNN is some-
 what less robust than the large DNN.

 Figure 8. Accuracy of two-target DoA estimation, with an allowable error of ±1°.

 3.2. Experiment Results
 One-target and two-target experiments are conducted with the experiment set-up
 presented in Sec. 2.3. In one-target scenarios, all the three methods (3D-CNN, DNN, and
 MUSIC) estimate DoA correctly with 1° precision. Fig. 9 shows the comparison of three
 NNs in terms of the accuracy of DoA estimation. The figure gives out three sets of accu-
 racy, i.e. with angular deviation of 0°, 2°, and 4°. Notice that MUSIC has been excluded

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 from the comparison; it is because that MUSIC often fails to identify two DoAs and is not
 qualified to compete with these methods. In two-target scenarios, with the tolerable an-
 gular deviation of 0°, the accuracy of 3D-CNN reaches 95.7%; while for the large DNN
 and DNN, the accuracy results are 93.7% and 89.1% respectively. When larger angular
 deviation is allowed, however, the accuracy of large DNN again goes to the top. This again
 shows that DNN has more robust performance.

 Figure 9. Accuracy of two-target DoA estimation vs. angular deviation.

 The 3D-CNN not only has better accuracy when zero angular deviation is allowed, it
 also excels in the critical cases where the angular difference between two targets is small.
 When the angular difference is less than 4°, the 3D-CNN has an accuracy of 97.8 %, while
 the DNN and the large DNN has 88.6% and 95.6% respectively.
 In summary, the 3D-CNN has the best performance, especially when the situation is
 critical. On the other hand, the slight decline in its performance when facing low-SNR
 data that is not experienced in the training phase shows that its performance, like many
 data-driven methods do, highly depend on the quality and quantity of the training data
 set available.

 4. Discussion
 DoA estimation is a classical problem for radar data processing. High-resolution
 methods such as MUSIC have been available for decades. Novel methods based on differ-
 ent types of NNs are widely regarded as the more promising approach for the old problem.
 In this paper we have demonstrated, as illustrated by Fig. 6-9, that two methods (3D-CNN
 and DNN) can consistently beat MUSIC with a large margin under both simulated and
 experiment scenarios. The 3D-CNN also has a better overall performance than the DNN,
 and it shows the potential of working with the RoI approach to generate high-resolution
 radar point clouds.
 It may be argued that the better performance of 3D-CNN is due to the fact that its
 input data-format (3-D tensor) retains more information from the raw radar data than the
 DNN’s input (covariance matrix) does. On the other hand, the stability of data benefitting
 from the averaging operation in estimating the covariance matrix gives a little advantage
 to the DNN, as demonstrated by the slightly more robust performance of DNN under
 some conditions.
 Another advantage of 3D-CNN is its better potential in future development. For in-
 stance, its input data contains the information along the slow time; therefore it is possible
 to train the same network for estimating target velocity, i.e. using the same 3D-CNN to
 output target’s range, velocity and DoA simultaneously is possible.
 Finally, not only the input data format plays an essential role in model learning, data
 augmentation is essential too. In acquiring the training data set, it is very difficult to con-
 duct experiments covering comprehensively the range of targets. The neural networks,

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 whether it is 3D-CNN or DNN, would perform poor if the range information in the train-
 ing set is not diverse enough. Eq. (10) offers a way to cover range diversity in doing data
 augmentation for preparing a comprehensive training data set.

 Author Contributions: Conceptualization, T.H.S. and F.T.C.; methodology, T.H.S. and F.T.C.; soft-
 ware, C.C.C.; investigation, all authors; data curation, B.S.W., C.C.C., and K.Y.T.; writing—original
 draft preparation, T.H.S., F.T.C., and B.S.W.; writing—review and editing, T.H.S. and F.T.C.; visu-
 alization, B.S.W., C.C.C., and K.Y.T.; supervision, T.H.S., F.T.C., and J.I.G.; project administration,
 T.H.S., F.T.C., and J.I.G.; funding acquisition, T.H.S., F.T.C., and J.I.G. All authors have read and
 agreed to the published version of the manuscript.
 Funding: This work is partially supported by the “Center for mmWave Smart Radar Systems and
 Technologies” under the “Featured Areas Research Center Program” within the framework of the
 Higher Education Sprout Project by the Ministry of Education (MOE), Taiwan, R.O.C., and partially
 supported under MOST projects with grants MOST 108-3017-F-009-001 and MOST 110-2634-F-009-
 028.
 Data Availability Statement: Training and testing data sets for the neural network models are avail-
 able at: https://drive.google.com/drive/u/1/folders/1JGfJP9RrxgR1IozXMxpdOmnaoUaq_vdC
 Acknowledgments: The authors would like to thank Jia-Jheng Lin and Jian-Xian Lu of Prof. Jiun-In
 Guo’s research group at Institute of Electronics, National Yang Ming Chiao Tung University
 (NCTU) for their generous support in so many aspects.
 Conflicts of Interest: The authors declare no conflict of interest.

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